Thursday, September 4, 2014

REPOST: Simple Pricing, the "stay-even" quantity

The marginal analysis of simple pricing, i.e., price at the point where MR=MC, is sometimes implemented by using a version of break-even analysis.  Instead of asking which price maximizes profit, you instead ask "will a given price increase, e.g., 5%, be profitable?"  To answer the question, we

1. Compute the stay-even quantity, the quantity you can afford to lose and still break even

2. Predict (or guess) whether the actual quantity lost will be greater or less than the stay-even quantity.

3.  If the actual quantity lost is less than the stay-even quantity, then the price increase will be profitable.   

Here is the derivation of the stay-even quantity:

The benefit of a price increase is the extra revenue you earn at the new (and lower) quantity, Benefit=dP*(Q+dQ)

The cost of a price increase is the margin on the lost sales, Cost=dQ(P-MC)

where dP=P1-P0, dQ=Q1-Q0, P0=initial price, P1=final price, Q0=initial quantity, and Q1=final quantity.

You compute the Quantity at which you are indifferent between raising price or not, and you get the formula:

(dQ/Q)=(dP/P)/[(dP/P)+m]

where m=(P-MC)/P, dQ/Q=% change in Q, and dP/P=% change in P.

EXAMPLE:

The stay even quantity for a 5% price increase for a firm with a 40% contribution margin is 11.1%=(5%)/[(5%)+(40%)].  If you expect to lose less than 11%, then a 5% price increase will be profitable.

REFERENCE:  Page 9 of
A Critical Analysis of Critical Loss Analysis,
Daniel P. O’Brien and Abraham L. Wickelgren, January 2003 (Published in Antitrust Law Journal) [PDF 236K]

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